Bring a large pot of lightly salted water to a boil. Cook lasagna noodles in the boiling water, stirring occasionally until tender yet firm to the bite, about 8 minutes. Drain.
Combine onion and garlic in a large skillet over medium-high heat. Cook and stir until onion is just translucent, about 5 minutes. Add turkey and cook until no longer pink, 5 to 7 minutes. Add mushrooms and season with salt and pepper; cook until softened, about 6 minutes. Remove from heat and allow to cool.
Wrap thawed spinach in a clean dish towel. Wring the spinach over a bowl to remove as much water as possible. Stir spinach, ricotta cheese, heavy cream, lemon zest, Italian seasoning, and nutmeg together in a separate bowl.
Preheat the oven to 375 degrees F (190 degrees C).
Coat the bottom of a 9x13-inch baking dish with 3 tablespoons spaghetti sauce. Layer 4 cooked lasagna noodles, 1/2 the spinach mixture, 1/2 the turkey mixture, 1/3 of the Asiago cheese, a third of the Cheddar cheese, and 3/4 cup spaghetti sauce on top. Layer 4 more noodles, the remaining spinach and turkey mixtures, another 1/3 of the Asiago cheese, another 1/3 of the Cheddar cheese, and another 3/4 cup spaghetti sauce. Top with the remaining 4 lasagna noodles, spaghetti sauce, Asiago cheese, and Cheddar cheese. Cover with aluminum foil.
Bake in the preheated oven for 35 minutes. Remove aluminum foil and increase oven temperature to 425 degrees F (220 degrees C). Continue baking until crust is golden, about 15 minutes more.